1
$\begingroup$

For a magnetic field along $\hat{x}$ the hamiltonian is given by: $H_{z}=V_x(c^{\dagger}_{\uparrow}c_{\downarrow}+c^{\dagger}_{\downarrow}c_{\uparrow})$.

If we follow the usual prescription of changing the spins: $\uparrow\quad \rightarrow \quad\downarrow$ and vice-versa, how do we see that the above hamiltonian breaks TR symmetry?

$\endgroup$
2
+50
$\begingroup$

Time reversal sends $c_\uparrow\rightarrow c_\downarrow$ and $c_\downarrow\rightarrow -c_\uparrow$. With this transformation, it should be clear the term is time odd.

See, e.g., here for more information. The key thing to realize is that while time reversal must send $k\rightarrow -k$ and $+z\rightarrow -z$, that still leaves the possibility of gaining an arbitrary phase under time reversal. To determine this phase, you demand that time reversal also sends $+x$ spins to $-x$ spins, and $+y$ spins to $-y$ spins. With these constraints, you can find the phase factors above.

$\endgroup$
  • $\begingroup$ Is there any reference for this? I thought the procedure only required flipping the spins. (and k -> -k in momentum space) $\endgroup$ – Arnab Barman Ray May 28 '18 at 16:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.